So I have heard that the distance between our star and moon is such that it is a cosmic wonder that it occludes our sun perfectly. I don't know 100% how accurate this simulation is but I no longer think that a total eclipse is all that rare.

Total solar eclipses are common, requiring only a moon whose angular size is larger than the star. It is far less common to have the ratio of angular sizes be a small range around unity, as it is with Earth/Luna/Sun at the present moment in their evolution. I can count on one hand the number of examples of these I've found in SpaceEngine (posted screenshots of one somewhere in the image dump thread; no idea what page anymore). Good job on finding another.

We have also to take into account the fact that the eccentricity of the orbit of the "moon" gives us a greater range of angular sizes and if that range includes the angular size of the star or overlaps in some parts of the range of angular sizes of the star (because the orbit of the planet may have it's own eccentricity) then there's a moment in history where perfect eclipses may occur.

Yes, but with larger eccentricity, "perfect" total eclipses will be rare for a given moon, requiring waiting through many alignments before one happens to be just above unity. It helps if the moon's inclination to the ecliptic is very small, so alignments happen with every orbit. For Luna/Sol the range of possible eclipse magnitudes is about 0.92 to 1.08, so there's about an equal mix of total eclipses to annular ones (slightly more annular since the moon spends more time closer to apoapsis).